Abstract
The automation of bargaining is receiving a lot of attention in artificial intelligence research. Indeed, considering that bargaining is the most common form of economic transaction, its automation could lead software agents to reach more effective agreements. In the present paper we focus on the best-known bargaining protocol, i.e., the alternating-offers protocol. It provides an elegant mechanism whereby a buyer and a seller can bilaterally bargain. Although this protocol and its refinements have been studied extensively, no work up to the present provides an adequate model for bargaining in electronic markets. A result of these settings means that multiple buyers are in competition with each other for the purchase of a good from the same seller while, analogously, multiple sellers are in competition with each other for the sale of a good to the same buyer. The study of these settings is of paramount importance, as they will be commonplace in real-world applications. In the present paper we provide a model that extends the alternating-offers protocol to include competition among agents.1 Our game theoretical analysis shows that the proposed model is satisfactory: it effectively captures the competition among agents, equilibrium strategies are efficiently computable, and the equilibrium outcome is unique. The main results we achieve are the following. 1) With m buyers and n sellers and when the outside option (i.e., the possibility of leaving a negotiation to start a new one) is inhibited, we show that it can be reduced to a problem of matching and that can be addressed by using the Gale-Shapley’s stable marriage algorithm. The equilibrium outcome is unique and can be computed in \(O(l \cdot m\cdot n \cdot \overline T + (m+n)^2)\), where l is the number of the issues and \(\overline{T}\) is the maximum length of the bargaining. 2) With m buyers and one seller and when the seller can exploit the outside option, we show that agents’ equilibrium strategies can be computed in \(O(l \cdot m \cdot \overline{T})\) and may be not unique. However, we show that a simple refinement of the agents’ utility functions leads to equilibrium uniqueness.
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References
Binmore, K., Shaked, A., Sutton, J.: An outside option expirement. Q. J. Econ. 104(4), 753–770 (1989)
Ceppi, S., Gatti, N.: An algorithmic game theory framework for solving bargaining with uncertainty. Tech. Rep. 2009.26, Politecnico di Milano, Dipartimento di Elettronica e Informazione (2009)
Chatterjee, K., Samuelson, L.: Bargaining under two-sided incomplete information: the unrestricted offers case. Oper. Res. 36(4), 605–618 (1988)
Cramton, P.C., Ausubel, L.M., Deneckere, R.J.: Handbook of Game Theory, vol. 3, pp. 1897–1945. Elsevier, Amsterdam (2002)
Dagan, N., Serrano, R., Volij, O.: Bargaining, coalitions and competition. Econ. Theory 15(2), 279–296 (2000)
Di Giunta, F., Gatti, N.: Alternating-offers bargaining under one-sided uncertainty on deadlines. In: Proceedings of the European Conference on Artificial Intelligence (ECAI), pp. 225–229. Riva del Garda, Italy (2006)
Di Giunta, F., Gatti, N.: Bargaining in-bundle over multiple issues in finite-horizon alternating-offers protocol. In: Proceedings of the Symposium on Artificial Intelligence and Mathematics (AIMATH). Fort Lauderdale, USA (2006)
Di Giunta, F., Gatti, N.: Bargaining over multiple issues in finite horizon alternating-offers protocol. Ann. Math. Artif. Intell. 47(3–4), 251–271 (2006)
Fatima, S.S., Wooldridge, M., Jennings, N.R.: An agenda-based framework for multi-issue negotiation. Artif. Intell. 152(1), 1–45 (2004)
Fatima, S.S., Wooldridge, M., Jennings, N.R.: On efficient procedures for multi-issue negotiation. In: Proceedings of Trading Agent Design and Analysis and Agent Mediated Electronic Commerce (TADA-AMEC) at the International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 71–84. Hakodate, Japan (2006)
Fatima, S.S., Wooldridge, M.J., Jennings, N.R.: Multi-issue negotiation with deadlines. J. Artif. Intell. Res. 27(1), 381–417 (2006)
Fudenberg, D., Levine, D.K., Tirole, J.: Incomplete information bargaining with outside opportunities. Q. J. Econ. 102(1), 37–50 (1987)
Fudenberg, D., Tirole, J.: Sequential bargaining with incomplete information. Rev. Econ. Stud. 50(2), 221–247 (1983)
Fudenberg, D., Tirole, J.: Game Theory. The MIT Press, Cambridge, USA (1991)
Gale, D.: Bargaining and competition. part I: Characterization. Econometrica 54(4), 785–806 (1986)
Gale, D.: Bargaining and competition. part II: existence. Econometrica 54(4), 807–818 (1986)
Gale, D.: Strategic Foundations of General Equilibrium—Dynamic Matching and Bargaining Games. Cambridge University Press, Cambridge (2000)
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–14 (1962)
Gatti, N.: Bargaining in markets with one-sided competition: model and analysis. In: Proceedings of the ACM/IEEE International Conference on Agent Intelligent Technologies (IAT), vol. IAT, pp. 443–446. Sydney, Australia (2008)
Gatti, N., Di Giunta, F., Marino, S.: Alternating-offers bargaining with one-sided uncertain deadlines: an efficient algorithm. Artif. Intell. 172(8–9), 1119–1157 (2008)
Gatti, N., Lazaric, A., Restelli, M.: Towards automated bargaining in electronic markets: a partially two-sided competition model. In: Proceedings of the Workshop on Agent-Mediated Electronic Commerce (AMEC) in the International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 29–42. Lisbon, Portugal (2008)
Gilpin, A., Sandholm, T.: Finding equilibria in large sequential games of imperfect information. In: Proceedings of the ACM Conference on Electronic Commerce (EC), pp. 160–169. ACM, New York (2006)
Halmos, P.R.: Measure Theory. Springer, Berlin (1974)
Harsanyi, J.C., Selten, R.: A generalized Nash solution for two-person bargaining games with incomplete information. Manage. Sci. 18, 80–106 (1972)
Jehiel, P., Moldovanu, P.: Cyclical delay in bargaining with externalities. Rev. Econ. Stud. 62(4), 619–637 (1995)
Jennings, N.R., Faratin, P., Lomuscio, A.R., Parsons, S., Sierra, C., Wooldridge, M.: Automated negotiation: prospects, methods, and challenges. International Journal of Group Decision and Negotiation 10(2), 199–215 (2001)
Koller, D., Megiddo, N., von Stengel, B.: Fast algorithms for finding randomized strategies in game trees. In: Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), pp. 750–759. ACM, New York (1994)
Koller, D., Pfeffer, A.: Representations and solutions for game-theoretic problem. Artif. Intell. 94(1), 167–215 (1997)
Kraus, S.: Strategic Negotiation in Multiagent Environments. MIT, Cambridge (2001)
Kreps, D.R., Wilson, R.: Sequential equilibria. Econometrica 50(4), 863–894 (1982)
Lazaric, A., Munoz de Cote, E., Gatti, N., Restelli, M.: Reinforcement learning in extensive form games with incomplete information: the bargaining case study. In: Proceedings of the International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 216–218. Honolulu, USA (2007)
Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York (1990)
Miltersen, P.B., Sorensen, T.B.: Computing sequential equilibria for two-player games. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithm, pp. 107–116. ACM, New York (2006)
Napel, S.: Bilateral Bargaining: Theory and Applications. Springer, Berlin (2002)
Nguyen, T.D., Jennings, N.R.: Coordinating multiple concurrent negotiations. In: Proceedings of the International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 1064–1071. New York, USA (2004)
Osborne, M.J., Rubinstein, A.: Bargaining and Markets. Academic, San Diego (1990)
Rosenschein, J.S., Zlotkin, G.: Rules of Encounter. Designing Conventions for Automated Negotiations among Computers. MIT, Cambridge (1994)
Rubinstein, A.: Perfect equilibrium in a bargaining model. Econometrica 50(1), 97–109 (1982)
Rubinstein, A.: A bargaining model with incomplete information about time preferences. Econometrica 53(5), 1151–1172 (1985)
Rubinstein, A., Wolinsky, A.: Equilibrium in a market with sequential bargaining. Econometrica 53(3), 1133–1150 (1985)
Rubinstein, A., Wolinsky, A.: Decentralized trading, strategic behavior and the walrasian outcome. Rev. Econ. Stud. 56(1), 63–78 (1990)
Sandholm, T.: Agents in electronic commerce: component technologies for automated negotiation and coalition formation. Autonomous Agents and Multi-Agent Systems 3(1), 73–96 (2000)
Sandholm, T., Vulkan, N.: Bargaining with deadlines. In: Proceedings of the American Association for Artificial Intelligence Conference (AAAI), pp. 44–51. Orlando, USA (1999)
Serrano, R.: The New Palgrave: a Dictionary of Economics, 2nd edn., chap. Bargaining. McMillian, London (2008)
Ståhl, I.: Bargaining Theory. Stockholm School of Economics, Stockholm (1972)
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Gatti, N. Extending the alternating-offers protocol in the presence of competition: models and theoretical analysis. Ann Math Artif Intell 55, 189 (2009). https://doi.org/10.1007/s10472-009-9158-1
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DOI: https://doi.org/10.1007/s10472-009-9158-1